On the semi-classical approximation to the wave function of the universe and its stochastic interpretation

Abstract

In quantum cosmology, a wave function ψ for a given theory can be obtained by solving the Wheeler-DeWitt equation, using the semi-classical approximation to the path integral over euclidean metrics to impose the boundary condition, as described by Hawking and his collaborators. If the universe is expanding as a quasi-de Sitter space-time, then it is possible to derive a Fokker-Planck equation for the probability distribution P, as shown by Starobinsky. Arguing by analogy with quantum mechanics in flat space-time, one would expect that P ∝ ψψ*. We examine this assertion by reference to the scale-invariant theory Image, whose wave function has been calculated in mini-superspace by Horowitz, and those classical solutions are de Sitter space-times. It appears that deviations from the relation P ∝ ψψ* are attributable to long-wavelength fluctuations δφe ≈ H/2π in the effective inflation field φe = √βR = √12β H. Their existence is taken into account in the derivation of the Fokker-Planck equation, but not in the derivation of ψ when this is restricted to mini-superspace. In the limit β → ∞, we find that δφe/φe → 0 and that P ∝ ψψ*. The scale invariant theory Image) can be similarly analyzed. Inclusion of a kinetic term Image destroys this similarly, which is restored however upon addition of a term Image.